I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while).
Specifically what I'm (pragmatically) interested in is the existence of some decomposition of a (co)vector field in a $3+1$ Lorentzian manifold into (gradient of a) scalar and non-scalar parts. I understand that this is not generically possible but I was reading this answer
and a few Einstein-Maxwell papers which imply that in some (not really very physically restrictive) restricted circumstances it is possible.
So my question is what fails in attempting direct generalisation to non-compact Lorentzian manifolds? With the follow up question: is global hyperbolicity enough of a restriction to allow such a generalisation?